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## #1 2013-05-13 13:40:52

dgiroux48
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### Multivariable Calculus Help

Having trouble even figuring out where to start on this problem. Our professor uploaded it on a kind of "cumulative final study guide", but I think we either use change of variables or surface integrals/parameterized surfaces (two topics we have covered). Let me know if you can help me tackle this one! Thanks!

2. We wish to build a fence on the landscape z = −1 − 8x^2 − 16y^2 from ground level up to height z = 0. The path of the fence is y=x^2 as x varies from x=−2 to x=+2. What will be the area of (the face of) the fence?

## #2 2013-05-14 13:18:52

John E. Franklin
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### Re: Multivariable Calculus Help

there is a formula for the length of a curve that might help.  jane fairfax gave it to me on this forum a few years ago.
search for length of curve or something...   then put that together with the height witha product i guess and maybe it'll work.

Imagine for a moment that even an earthworm may possess a love of self and a love of others.

## #3 2013-05-14 13:32:46

John E. Franklin
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### Re: Multivariable Calculus Help

I searched for Jane's formula but couldn't find it.
but i did find this weird discussion of mine and carlos.
http://www.mathisfunforum.com/viewtopic.php?id=2168

Imagine for a moment that even an earthworm may possess a love of self and a love of others.

## #4 2013-05-21 05:29:54

dumbcow
Guest

### Re: Multivariable Calculus Help

The surface integral would give the area of a specified surfaced of the paraboloid (z) but we want area of the fence which comes up from surface to z=0 along the points y=x^2.

The area of fence can be found by multiplying height by length of fence dictated by path (y=x^2) along surface

Area = integral [-2,2] |z| dL

dL^2 = dx^2 + dy^2 +dz^2
hence

dL = sqrt(1 + (dy/dx)^2 + (dz/dx)^2 ) dx

since path of fence is y=x^2, we can put z in terms of x
z = -1 -8x^2 -16x^4

dy/dx = 2x
dz/dx = -16x -64x^3

substituting into Area integral:

Area = integral[-2,2] (1+8x^2+16x^4) sqrt(1+260x^2+2048x^4 +4096x^6) dx